3/x+2 = 2/x^2+3x - 4/x

Question

solve for x

3/x+2 = 2/x^2+3x - 4/x

Answer 1

3/x+2 = 2/x^2+3x - 4/x

multiply all terms by x to get

3x +2x^2 = 2 +3x^3 -4x

combine like terms

3x^3 + x^2 -7x + 2 = 0

x f(x)

-2 -7

-1 7

0 2

1 -1

2 16

there are 3 real solutions, 2 positive, 1 negative by Descartes' Method

one negative is between -1 and -2, maybe about - 1.5

one positive is between 0 and 1

another positive is between 1 and 16

both positive solutions are close to x=1, one a little above, the other a little below

there are no integer solutions

UNLESS you really meant to write

3/(x+2) = 2/(x^2+3x) - 4/x

then multiply by LCD to eliminate fractions

3x(x +3) = 2(x+2) - 4(x+2)(x+3)

expand and combine like terms

3x^2 + 9x = 2x +4 -4x^2 -20x -24

3x^2 +7x -4 = -4x^2 -20x -24

7x^2 +27x +20 = 0

(7x+20)(x+1) = 0

x =-1 and -20/7